Introduction to Focus Issue: Synchronization in Complex Networks
Johan A. K. Suykens1and Grigory V. Osipov2
1K.U. Leuven, ESAT-SCD/SISTA, Kasteelpark Arenberg 10, B-3001 Leuven (Heverlee), Belgium
2Department of Control Theory, Nizhny Novgorod University, 23, Gagarin Avenue, 603950 Nizhny Novgorod, Russia
共Received 28 August 2008; accepted 28 August 2008; published online 22 September 2008兲 Synchronization in large ensembles of coupled interacting units is a fundamental phenomenon relevant for the understanding of working mechanisms in neuronal networks, genetic networks, coupled electrical and laser networks, coupled mechanical systems, networks in social sciences, and others. It relates to mathematical and computational analysis of the existence of different states and its stability, clustering, bifurcations and chaos, robustness and sensitivity analysis, etc., at the intersection between synchronization and pattern formation in complex networks. This interdisci-plinary oriented Focus Issue presents recent progress in this area with contributions on generic methods, specific model studies, and applications. © 2008 American Institute of Physics.
关DOI:10.1063/1.2985139兴
Nowadays, the main focus in the study of synchronization phenomena is moving from the analysis of a low number of interacting units to systems consisting of thousands and even millions of nodes. Such systems are ubiquitous in nature and engineering. The understanding of com-mon properties of network synchronization, mechanisms of its appearance, and strategies of its control in depen-dence on dynamics of individual nodes, networks archi-tecture, and internode coupling types is a subject of cur-rent research.
I. CONTEXT
Complex networks are systems composed of a 共very兲 large number of nodes often characterized by a complex to-pology and possibly complex types of interactions.1 The structure of a network is typically characterized in a graph theoretic way. Indicators, such as, degree distributions, aver-age path length, and others have been used to qualitatively distinguish between different types of complex networks.2,3 In general, one may have different network topologies, such as, according to regular networks共e.g., a ring structure, near-est neighbor coupling, globally connected兲, random graphs, small world networks, and scale free networks.1,4The type of coupling can be linear or nonlinear, constant or changing in time, etc. The dynamics of the individual nodes might in general be regular or chaotic.
The formation of collective behavior in such large net-works of coupled dynamical elements is one of the new problems in the study of dynamical systems. It poses many challenges for achieving a better theoretical understanding as well as for applications in various disciplines, ranging from physics, chemistry, earth sciences via biology and neuro-science to engineering, business, and social neuro-sciences.
Due to the large number of effective degrees of freedom in spatially extended systems, a rich variety of spatio-temporal regimes is observed. Three main types of collective behavior are typically distinguished: a fully incoherent state
共or highly developed spatio-temporal disorder兲, partially co-herent states 共where some of the elements in the network behave according to a common rhythm兲 or a fully coherent state共a regime of globally synchronized elements兲. The basic phenomenon of these structure formations is synchroniza-tion, i.e., a regime of coherent activity, which is universal in many dynamical systems and can be understood from the analysis of common models of oscillatory networks.
The subject of synchronization of chaotic systems has been intensively investigated since the work by Pecora and Carroll,5however, the main focus in the past has been on low dimensional systems and regular network topologies. Syn-chronization in complex networks poses new challenges to-wards the understanding of synchrony and its formation in large scale networks, also for more complex network topolo-gies.
Different synchronization phenomena have been widely studied and reported in the literature. A number of related book references that address this subject are the following. The book by Pikovsky et al.6 overviews many currently known synchronization phenomena 共including tion of a periodic oscillator by external force, synchroniza-tion of chaotic systems, synchronizasynchroniza-tion in the presence of noise and in oscillatory media兲. An introduction to the sub-ject of synchronization 共without formulas兲 illustrated with many biological and technological examples has been given in the book by Strogatz.7A popular introduction to synchro-nization theory illustrated by many examples can be found in Refs.8and9. Synchronization by external force, synchroni-zation of oscillators, and the influence of noise on synchro-nization have been studied in some parts of the monograph.10 Kuramoto11 studied synchronization in large populations of phase oscillators and in continuous media. Synchronization from the viewpoint of biological applications is studied in Ref. 12, and synchronization from an engineering perspec-tive in Refs. 13–16. Blekhman17has addressed the synchro-nization in mechanical oscillators, electronic, and quantum generators, etc. Noisy influenced synchronization in periodic CHAOS 18, 037101共2008兲
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systems was developed by Stratonovich.18 Books that are entirely related to synchronization problems in networks of coupled oscillators are Afraimovich et al.19共focusing on net-works of identical phase locked loops兲, Manrubia et al.,20 and recently Osipov et al.21
II. THIS FOCUS ISSUE
The contributions to this Focus Issue on Synchronization in Complex Networks are grouped into three parts:
• Generic methods • Specific models • Applications
The first part on Generic methods addresses problems such as: What is the relation between the topology of the network and synchronization into clusters? How could one design the topology of a network as to achieve desired clus-ters? Under which conditions is synchronization still achieved in the case of time delays or communication con-straints? How can one enhance the synchronizability of com-plex networks and rewire the network? Does adding nodes increase synchronizability? How can one efficiently control time-varying couplings?
The second part contains in-depth studies of Specific models. It includes synchronization in networks of, e.g., Kuramoto models, Hodgkin–Huxley neuron models, integrate-and-fire oscillators, and coupled map lattices.
The third part on Applications contains contributions on synchronization of genetic oscillators, synchronization of cells undergoing metabolic oscillations, synchronization in ciliary arrays, collective phenomena in cardiac cell cultures, phase synchrony from multielectrode arrays visual cortex measurements, and synchrony in electrocorticographic sig-nals.
A brief summary of the contributions is given here for each of the three parts.
A. Generic methods
Chen and Duan22 address the fundamental problem of complex network synchronizability from a graph-theoretic view. They study the relationship between network synchro-nizability and structural network parameters, such as, aver-age distance and degree distribution. The effect of adding new edges to the network with respect to the network syn-chronizability is investigated.
Wu23 looks at pinning control of a set of coupled dy-namical systems forced by an external signal. Suitable loca-tions of pinning are investigated and how it relates to the network topology.
In Ref. 24, Rad et al. study efficient rewirings, i.e., eliminating an edge and creating a new edge elsewhere, in order to achieve enhanced synchronizability. Different target functions for optimization are explored including the eigen-ratio of the Laplacian.
Hagberg and Schult25 study synchronization of diffu-sively coupled identical oscillators and examine the effect of adding and removing edges on the network synchronization.
Through the master stability function approach a separation of the network topology from the consideration of the dy-namics of the oscillators is obtained.
V. Belykh et al.26 study conditions for cluster partition-ing into ensembles for the case of identical chaotic systems. A distinction is made between conditional and unconditional clusters. The design problem of organizing clusters into a given configuration is discussed.
In Ref.27, Pogromsky investigates partial synchroniza-tion in networks of linearly coupled oscillators for the case of symmetry in the couplings.
Oguchi et al.28consider synchronization of a set of iden-tical nonlinear systems unidirectionally or bidirectionally coupled with time delay. Sufficient conditions for synchroni-zation are obtained in terms of linear matrix inequalities based on a Lyapunov–Krasovskii function approach.
Fradkov et al.29 study synchronization of nonlinear sys-tems under information constraints with a limited informa-tion capacity of the coupling channel. An upper bound on the limit synchronization error is derived as a function of the transmission error. This is analyzed with respect to different network topologies.
In Ref.30, De Lellis et al. discuss adaptive strategies for the synchronization of complex networks with time-varying coupling. These include vertex-based and edge-based ap-proaches for which global synchronization is analyzed.
Y. Wu et al.31 study complete synchronization in small world networks of identical oscillators. Synchronizability of the network is analyzed with respect to a time-varying cou-pling.
B. Specific models
Antonsen et al.32address large systems of coupled oscil-lators subjected to a periodic external drive for equal strength, all-to-all sine coupling of phase oscillators. Station-ary states and their stability are determined. With sufficient drive the system produces regular rhythms.
Ott and Antonsen33 show that in the infinite size limit certain systems of globally coupled phase oscillators display low dimensional dynamics. An exact closed form solution for the nonlinear time evolution of the Kuramoto problem is obtained. Also for extensions to this model low dimensional behavior is demonstrated.
In Ref.34, So et al. study a network of two large inter-acting heterogeneous populations of limit-cycle oscillators under a switching connectivity. For sufficiently high switch-ing frequency the system behaves as if the connectivity were static.
Ott et al.35 show the existence of echo phenomena in large systems of coupled oscillators below the transition to phase coherence. This is exemplified by the Kuramoto model.
D’Huys et al.36 address the effect of coupling delays on the synchronization properties of network motifs. The influ-ence of coupling delay and symmetry on synchronization patterns in uni- and bidirectionally coupled rings and open chains of Kuramoto oscillators are analyzed.
037101-2 J. A. K. Suykens and G. V. Osipov Chaos 18, 037101共2008兲
In Ref.37, Wang et al. show analytically for a general-ized Kuramoto model, how applying gradient coupling can enhance the synchronization in nonidentical scale-free net-works.
Ahnert and Pikovsky38 characterize different types of traveling waves in a chain of dispersively coupled phase os-cillators. Stability of the waves is studied.
Nowotny et al.39 discuss neural synchrony including synchronization with activity dependent coupling, synchroni-zation of bursts generated by a group of nonsymmetrically coupled inhibitory neurons, partial synchronization of small composite structures, and synchronization on a mesoscopic scale.
In Ref.40, Shilnikov et al. study polyrhythmic synchro-nization in bursting networking motifs. A universal burst-duration based mechanism of the emergence of synchronous bursting patterns is described for larger networks, such as, pattern generators that control locomotion activities.
Komarov et al.41 study collective phenomena in small and large arrays of nonidentical cells coupled by models of electrical and chemical synapses and a Hodgkin–Huxley-type model. Various regimes of phase synchronization are observed.
Mauroy and Sepulchre42investigate clustering behavior in a model of integrate-and-fire oscillators with excitory pulse coupling. Global convergence to phase-locked cluster behavior is proven for identical oscillators. The robustness of the clustering behavior is addressed for nonidentical oscilla-tors.
In Ref.43, Anishchenko et al. address synchronization of quasiperiodic oscillations by an external harmonic signal. Synchronization of a resonant limit cycle on a two-dimensional torus is studied.
Zhusubaliyev and Mosekilde44investigate the formation and destruction of multilayered tori in coupled map systems. Torus destruction involves nonlocal bifurcations and repre-sents a transition to chaos scenario. The tori formation is related to noninvertibility of the maps.
Cencini et al.45 study chaotic synchronizations of spa-tially extended systems of nonequilibrium phase transitions. Short- and long-range interactions are addressed in two rep-licas of coupled map lattices with a coupling decaying as a power law.
C. Applications
Zhou et al.46 address synchronization of genetic oscilla-tors. It is shown how a collective rhythm is achieved through synchronization-induced intercellular communication and how an ensemble is synchronized by a common noisy sig-naling molecule.
Gonze et al.47 study in-phase and out-phase synchroni-zation in a model based on global coupling of cells undergo-ing metabolic oscillations.
In Ref.48, Niedermayer et al. investigate synchroniza-tion, phase locking, and metachronal wave formation in cili-ary chains. Beating cilia are described as phase oscillators moving on circular trajectories with variable radius, where the radial degree of freedom enables hydrodynamically in-duced synchronization.
Kryukov et al.49 study collective phenomena in hetero-geneous cardiac cell cultures consisting of a mixture of pas-sive, oscillatory, and excitable cells. For oscillatory and ex-citable media, transitions from incoherent behavior to partially coherent behavior and finally to global synchroni-zation are discussed.
Manyakov and Van Hulle50 investigate phase synchrony with respect to local field potentials recorded with a multi-electrode array implanted in the monkey visual cortex. An information-theoretic measure is used to characterize the phase synchrony based on normalized mutual information between pairs of local field potential phases.
In Ref. 51, Kozma and Freeman address intermittent spatio-temporal desynchronization and sequenced synchrony in electrocorticographic signals from the brain surface. Fil-tered signals confirm the presence of a shared mean fre-quency in a frame of synchronized oscillation.
ACKNOWLEDGMENTS
Guest editor Johan Suykens acknowledges support from K. U. Leuven, the Flemish government, FWO, and the Bel-gian federal science policy office共FWO G.0226.06, CoE EF/ 05/006, GOA AMBioRICS, IUAP DYSCO, BIL/05/43兲. Guest editor Grigory Osipov acknowledges financial support from RFBR-NSC Project No. 05-02-90567, and RFBR Project Nos. 05-02-19815, 06-02-16596, 08-02-97049. We are grateful to Jurgen Kurths, Vladimir Belykh, and Joos Vandewalle. We express our special thanks to Janis Bennett, assistant editor of this journal, for her continuous help and assistance in preparing this focus issue. Finally, we appreci-ate the efforts of the many reviewers who helped to improve the quality of the contributions for this focus issue.
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